notes/uni/mmme/2046_dynamics_and_control/control.md

---
author: Akbar Rahman
date: \today
title: MMME2046 // Control
tags: [ mmme2046, uon, uni, control ]
uuid: 73e04dd2-ee4c-4952-a9b7-7df3930d2d2d
lecture_slides: [ ./lecture_slides/Control 1 2023.pdf, ./lecture_slides/Control 2 2022.pdf, ./lecture_slides/Control Lecture 3 2022.pptx ]
exercise_sheets: [ ./exercise_sheets/control.pdf, ./exercise_sheets/control_sols_odd.pdf ]
---

# Errata

## Exercise Sheets

### ES1, Q5 (p3)

Output column on row 3c should be $h_2$ not $h_3$.

## Lecture Slides 2 p26

First line should be

$$C(s) = \frac{5}{s(s+5)} = \frac 1s \frac{1}{1+0.2s}$$

# System and Block Diagrams

# Laplace Transform

$$F(s) = \mathscr L {F(t)} = \int^\infty_0 f(t)e^{-st} \mathrm{d}t$$

where $s = \alpha + j\omega$

The function $F(s)$ is often much easier to manipulate than periodic function $f(t)$.

## Final Value Theorem

As $f(t)$ tends to infinity, $sF(s)$ tends to 0.

## Example

$$\dot x_o = ax_o = ax_i$$

where $x_o$ is the output and $x_i$ is the input

Take the Laplace transform:

$$sX_o(s) + aX_o(s) = aX_i(s)$$

Rearrange to get equation for the transfer function:

$$G(s) = \frac{X_o}{X_i} = \frac{a}{s+a}$$

$$ X_o = GX_i $$

If $X_i$ is a unit step, then:

$$X_i = \frac1s$$

and

$$X_o = \frac{a}{s(s+a)}$$

Taking the inverse gives:

$$X_0 = 1 - e^{-at}$$

# Non-Linearity

Sometimes, components of a system will not reduce to a simple linear relationship.
When this is the case superposition and Laplace transforms do not apply/are not valid.

Reasons for this include:

- saturation

  ![](./images/vimscrot-2023-02-06T16:10:06,638264779+00:00.png)

- backlash 

  ![](./images/vimscrot-2023-02-06T16:10:23,750576923+00:00.png)

- clearance 

  ![](./images/vimscrot-2023-02-06T16:10:39,624151288+00:00.png)

- coulomb friction 

  ![](./images/vimscrot-2023-02-06T16:10:55,163385436+00:00.png)

- material non-linearity

  ![](./images/vimscrot-2023-02-06T16:11:17,999306580+00:00.png)

- flow through an orifice  (choked flow)

  ![](./images/vimscrot-2023-02-06T16:11:34,160399051+00:00.png)

## Linearisation

System behaviour is approximated to a linear relationship near the "nominal" operating point:

![](./images/vimscrot-2023-02-06T16:13:20,353784072+00:00.png)